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We use a variational approach to study existence and regularity of solutionsfor a Neumann $p$-Laplacian problem with a reaction term on metric spacesequipped with a doubling measure and supporting a Poincar\'e inequality. Tracetheorems for functions with bounded variation are applied in the definition ofthe variational functional and minimizers are shown to satisfy De Giorgi typeconditions.

Neumann $p$-Laplacian problems with a reaction term on metric spaces